Basically limbo was an idea thrown around by medieval theologians who favored a somewhat legalistic interpretation of the Holy Scriptures. The idea was:
A) You must be baptized to get into heaven.
B) But some infants die before they can be baptized.
C) So these unbaptized infants can't go to heaven.
D) But if God is just, He wouldn't send innocent infants to hell.
E) But because they're unbaptized, even though they won't be sent to hell, they can't enter heaven either. Ergo:
F) They go to a totally different place where they live happily ever after, and for lack of a better term, we'll call that place "limbo".
Limbo was never embraced as an official doctrine of the Church; it was merely a concept, an idea, that the Scholastic theologians contemplated. Today, the concept has been shelved, and as the current
Catechism states, unbaptized infants are left to the compassion and mercy of God, Who will surely care for them and place them where He sees fit, in His infinite wisdom.
If you feel like wading through the article, here's the
Catholic Encyclopedia article on limbo:
http://www.newadvent.org/cathen/09256a.htm
And here's the passage from the current
Catechism:
THE CATECHISM OF THE CATHOLIC CHURCH
Part 2, Section 2, Chapter 1, Article 1, SubSection 6:
1261 As regards children who have died without Baptism, the Church can only entrust them to the mercy of God, as she does in her funeral rites for them. Indeed, the great mercy of God who desires that all men should be saved, and Jesus' tenderness toward children which caused him to say: "Let the children come to me, do not hinder them," 64 allow us to hope that there is a way of salvation for children who have died without Baptism. All the more urgent is the Church's call not to prevent little children coming to Christ through the gift of holy Baptism.